Start Free Trial

Cancel Anytime.

D is a f

D is a fixed point so that the point B may only move vertical to the member BD. B moves from B to ixed point so that the point B may only move vertical to the member BD. B moves from B to B’. In aB’. In a similar way E is a f

similar way E is a f ixed point and C can only move vertically to the member to C’. A may move horizontally.ixed point and C can only move vertically to the member to C’. A may move horizontally.

If one looks at the member AB both ends may move so we will find a momentary centre of rotation O

If one looks at the member AB both ends may move so we will find a momentary centre of rotation O22verticalvertical to the direction of

to the direction of movemenmovement. Both ends of t. Both ends of member BC can move so we member BC can move so we will find a will find a momentary centre of momentary centre of  rotation, O

rotation, O11vertical to the direction of movement of B and C.vertical to the direction of movement of B and C.

Because movements are small relative to the length of the member, the tan

Because movements are small relative to the length of the member, the tan

ψ  ψ 

= the angle= the angle

ψ  ψ 

^..

BB’ = 5 x

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

Structure with sway

Structure with sway mechanismmechanism..

Determine the number of independent sway mechanisms:

Determine the number of independent sway mechanisms:

s

s = = 44 r r = = 55 s s + + r r = = 99 n

n = = 55 22n n = = 1100 2n –(s + r) = 1 2n –(s + r) = 1

1 independent sway mechanism!

1 independent sway mechanism!

Unknowns

Unknowns

θθ

AA use modified slope-deflection equationuse modified slope-deflection equation

θθ

^{BB ??}

θθ

CC use modified slope-deflection equationuse modified slope-deflection equation

θθ

^{BB 00}

θθ

EE 00

ψ 

ψ 

^??

2 unknown – we require 2 equations 2 unknown – we require 2 equations Set

Set

ψ  ψ 

DBDB = -= -

ψ  ψ 

BB’ = 4 BB’ = 4

ψ  ψ 

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

( ( ))

( ( ))

2

2 22 33 4

BD 4

BD BB

EI  M  EI 

M 

= = ⋅⋅ ⋅ ⋅ −

θ θ

− ⋅ ⋅ −−

ψ ψ  1

1,,00 11,,55

BD

BD BB

M

M

= = ⋅ ⋅ ⋅ + ⋅ ⋅

EEII

⋅ + ⋅ ⋅

θ θ EEII ψ ψ     0

0 22 11,,2255 3333,,7755 00

B

B BB

M

M

= = ∴ ∴ ⋅ ⋅ ⋅

EEII

⋅ +

θ θ

+ ⋅ ⋅ ⋅

EEII

⋅ −

ψ ψ 

−

  

==

∑

∑

⁽¹⁾⁽¹⁾

To determine the second equation one must view all the external forces on the structure:

To determine the second equation one must view all the external forces on the structure:

As it is difficult to

As it is difficult to determine YDB and YEC we will take moments about a point where their momedetermine YDB and YEC we will take moments about a point where their moment is knownnt is known to be 0.

to be 0. The momentary centre of rotation, OThe momentary centre of rotation, O11, is such a point., is such a point.

Determine the unknown forces in terms of the unknown rotations and translational angles.

Determine the unknown forces in terms of the unknown rotations and translational angles.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Take moments about the momentary centre of rotation:

Take moments about the momentary centre of rotation:

V

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

We have three unknowns, namely

We have three unknowns, namely

θθ

BB,,

θθ

DD andand

ψ  ψ 

. We require three equations to solve . We require three equations to solve these unknowns.these unknowns.

=

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Third equation may be obtained from the vertical equilibrium of node C Third equation may be obtained from the vertical equilibrium of node C

-V

Solve the three simultaneous equations:

Solve the three simultaneous equations:

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The support D of the structure undergoes the following displacement, 10 mm vertically down and 20 mm The support D of the structure undergoes the following displacement, 10 mm vertically down and 20 mm horizontally to the left. E = 200 GPa and I = 50 x 10

horizontally to the left. E = 200 GPa and I = 50 x 10^-6-6mm⁴⁴..

If one determines the number of independent sway mechanisms we see that there is one. The unknowns are If one determines the number of independent sway mechanisms we see that there is one. The unknowns are thus

thus

θθ

^BBandand

ψ  ψ 

^..

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The total sway as

The total sway as a result of a result of the displacementhe displacements is ts is the sum of the the sum of the individual sway angles. Therefore:individual sway angles. Therefore:

3

3 3 3 33

2

2,,55 1100 33,,7755 1100 66,,2255 1100

 AB

 AB xx xx xx   

ψ 

ψ 

= = − −

^{− −}

− −

^{− −}

= = −−

⁻⁻

3

3 3 3 33

1

1,,11111111 1100 11,,66666677 1100 22,,77777788 1100

BC 

BC  xx xx xx   

ψ 

ψ 

= =

^{− −}

+ +

^{− −}

==

⁻⁻

ψ 

ψ BDBD= 5,0 x 10= 5,0 x 10^-3-3

To determine the relative sway angles:

To determine the relative sway angles:

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

For the second equation one must look at all the external forces that are applied to the structure.

For the second equation one must look at all the external forces that are applied to the structure.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

The world’s largest digital library

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions!

Start Free Trial

Cancel Anytime.

M

MBCBC = + 58,418 kN.m= + 58,418 kN.m M

MBDBD = - 77,990 kN.m= - 77,990 kN.m M

MDBDB = - 61,804 kN.m= - 61,804 kN.m

In document Slope Deflection Examples (Page 21-36)